4G3 Computational Neuroscience Assignment

Assignment Task

Network Dynamics

This coursework is a basic study of how well a recurrent network model of primary visual cortex (V1) can represent a specific feature (in this case, the orientation) of a visual stimulus, in the face of readout noise, depending on its connectivity. The basic setup you will investigate is depicted in the following diagram:

In broad strokes, the stimulus orientation θ is encoded into V1 activity, which is then decoded into an estimate ˆθ which we hope to be as close as possible to θ. Noise is injected in the output, thus corrupting the representation of θ and giving rise to reconstruction errors (ˆθ ̸= θ). The question we ask here is: how does the structure of recurrent connectivity in the V1 network affect the reconstruction error?

Details

The default values of the various parameters used in the following are given in Table 1. Let {ϕi = 2πi m } be a grid of m regularly spaced orientations1 . The stimulus orientation θ, also measured in radians between 0 and 2π, is first encoded into a vector h(θ) ∈ R m with the i th element given by

hi(θ) = V(ϕi − θ) with V(z) = exp ( cos(z) − 1/κ 2)

This vector is then used as a constant input for the dynamics of V12 :

τ dr /dt = −r + Wr + Bh(θ)

where r(t) ∈ R n is a vector whose i th element ri(t) represents the momentary firing rate of V1 neuron i (relative to some positive baseline), B ∈ R n×m is a matrix of feedforward input weights, W is a matrix of recurrent connection strengths, and τ is the characteristic neuronal time constant. You may systematically set the initial condition r(0) to 0.

A noisy readout of V1 activity can be obtained at any time t in the form

o˜(t) = Cr(t) + σε(t)

where C ∈ R m×n is a matrix of output weights and each element of ε(t) is drawn from N (0, 1) independently (across vector elements, time, and trials). Finally, this noisy readout is decoded into a momentary estimate of θ according to3

θ(t) = atan (Σ i o˜i(t) sin ϕi / Σi o˜i(t) cos ϕi )

1. Integrate the dynamics of each of the 4 models with their default parameters for θ = π, and show the corresponding response r(t) at t = {10, 50, 100, 200} ms.

2. The response of Model should look a lot noisier across the V1 population than in the other models. By using a combination of analytical derivations (including the eigendecomposition of the W(2) matrix) and any additional numerics you deem useful, provide a rational explanation for this phenomenon.

3. You should also find that Model responds more strongly than the other models. By using a combination of analytical derivations (including the eigendecomposition of the W(3) matrix) and any additional numerics you deem useful, provide a rational explanation for this phenomenon.

4. Models respond more slowly than Models 1 and 4. Based on your answers to the two previous questions, provide a rational explanation for this phenomenon.

5. For each model, compute and show the time course of the corresponding decoding accuracy, averaged over repeated trials, with θ drawn uniformly in each trial. Based on your answers so far, explain your decoding accuracy results and in particular how accuracy differs across models. 

6. Revisit the behaviour of Model 4, now setting α ′ = 5 (instead of 0.9 previously). You should find that the reconstruction of θ is now the best amongst the four models, both in terms of speed and accuracy. Explain this phenomenon.

7. What happens in Models 2 and 3 if you similarly set α = 5? (instead of 0.9 previously). Why?

8. How could Model 4’s input weights B be chosen differently (from Equation 5) to further improve reconstruction accuracy in that model? Justify your suggestion and try it out.

9. How could Model 4’s output weights C be chosen differently (from Equation 5) to further improve reconstruction accuracy in that model? Justify your suggestion and try it out.

10. It appears that the balanced model can be made arbitrarily good for coding speed and fidelity, by increasing α . Where is the catch?

11. Briefly conclude by discussing the biological implications of your results, in particular in regard to the putative role of excitation-inhibition balance in the speed and precision of orientation coding in V1.

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